Influence of depth-dependent Brinkman viscosity on the onset of convection in ternary porous layers

Transcription

1 Influence of depth-dependent Brinkman viscosity on the onset of convection in ternary porous layers Salvatore Rionero University of Naples Federico II. Department of Mathematics And Applications R.Caccioppoli. Via Cinzia Italy Workshop on PDE s and Biomedical Applications December 4-6, 2014, Lisbon, Portugal 1

2 β = T 1 T 2, β = β C buoyancy force=weight. d 2

3 Influence of depth-dependent Brinkman viscosity on the onset of convection in ternary porous layers Index 1) Introduction of the physical phenomenon; 2) References and Discussion; 3) Mathematical preliminaries; 4) The effective unknown fields; 5) Linear instability; 6) Nonlinear stability via the Auxiliary System Method; 7) Global non linear stability via symmetries and skew-symmetries hidden in the model; 8) Applications. 3

4 1 Introduction Because of its great relevance in geophysical phenomena and in planning new artificial porous materials, the onset of convection in porous layers with variable permeability and/or viscosities, in the past as nowadays, has attracted the attention of scientists { see [1]- [13], [28] and the references therein }. As concerns the geophysical phenomena, we confine ourselves to mention the increase in viscosity with depth in the earth s mantle [1]; the permeability changes due to mineral diagenesis in fractured crust [7]; the porosity changes due to subterranean movements, the increase in permeability and 4

5 porosity near solid wall {see [10]-[11] and references therein}. As concerns the stratification of porosity in artificial porous materials, we recall that for insulating purposes, the porosity has to be stratified in such a way to delay or inhibit heat transfer and hence in such a way to produce an high thermal critical Rayleigh number. On the contrary, when rapid heat transfer is requested (such as in cooling pipes used in modern devices), the porosity has to be stratified in such a way to produce low thermal critical Rayleigh numbers. In the present paper, in the Darcy-Boussinesq-Brinkman scheme, the onset of convection in a porous horizontal layer L with depthdependent permeability and viscosities - via the Auxiliary System Method [17]-[19] - is investigated 1. Section 2 is dedicated to some preliminaries and to the functional spaces in which the problem is embedded. In section 3 the main relation between the effective unknown fields is obtained. The linear instability is studied in the subsequent section 4. The nonlin- 1 In [28] the onset of convection in ternary porous layers with depth-dependent permeability and viscosity has been studied in the absence of Brinkman term. 5

6 ear stability is considered in section 5. In section 6, looking for symmetries and skew-symmetries hidden in the Darcy-Boussinesq- Brinkman model, for classes of values of the Prandtl numbers, a condition in closed form guaranteeing the global nonlinear stability, is furnished. Section 7 is devoted to the applications. References [1] Torrance, K.E.; Turcotte, D.L. Thermal convection with large viscosity variations. J. Fluid Mech. 47, , (1971). [2] Kassoy, D.R.; Zebib, A. Variable viscosity effects on the onset of convection in porous media. Phys. Fluids 18, , (1975). [3] Straughan, B. Stability Criteria for convection with large viscosity variations. Acta Mech. 61, 59-72, (1986). 6

7 [4] McKibbin, R. Heat transfer in a Vertically-layered porous medium heated from below. Transp. Porous Med. 1, , (1986). [5] Rosenberg, N.J., Spera, F.J. Role of anisotropic and/or layered permeability in hydrothermal system. Geophys. Res. Lett. 17, , (1990). [6] Rees, D.A.S., Pop, I. Vertical free convection in a porous medium with variable permeability effects. Int. J. Heat Mass Transfer, 43, , (2000). [7] Fontaine, F. Jh., Rabinowicz, M., Boulegue, J. Permeability changes due to mineral diagenesis in fractured crust. Earth and Planetary Science Letters, 184, , (2001). [8] Nield, D.A., Kuznetsov, A.V. The effect of a transition layer between a fluid and a porous medium: shear flow in a channel. Transp. Porous Med., 72, , (2009). 7

8 [9] Alloui, Z., Bennacer, R., Vasseur, P. Variable Permeability effect on convection in binary mixtures saturating a porous layer. Heat and Mass Transfer, 45, , (2009). [10] Hamdan, M.H., Kamel, M.T., Flow through variable permeability porous layers. Adv. Theor. Appl. Mech., 4, n.3, , (2011). [11] Hamdan, M.H., Kamel, M.T., Siyyam, H.I. A permeability function for Brinkman s equation. Proceedings of 11th WSEAS Int. Conf. on Mathematical Methods, Computational Techniques and intelligent systems. (2009) [12] Rionero, S. Onset of convection in porous materials with vertically stratified porosity. Acta Mech., 222, , (2011). [13] Nield, D.A., Kuznetsov, A.V. Optimization of forced convection heat transfer in a composite porous medium channel. Transp. Porous Med., 99, , (2013). 8

9 [14] Nield, D.A., Bejan, A. Conduction in Porous Media, 4th edition, Springer, (2013). [15] Straughan, B. Stability and wave motion in porous media. Springer Appl. Math. Sc. 165, (2008). [16] Flavin, J.N.; Rionero, S., Qualitative estimates for partial differential equations: an introduction. Boca Raton (FL): CRC Press, (1996). [17] Rionero, S. Absence of subcritical instabilities and global non linear stability for porous ternary diffusive-convective fluid mixtures. Phys. Fluids, 24, issue 10, , (2012), 17 p. [18] Rionero, S. Multicomponent diffusive-convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities and global non linear stability for any number of salts. Phys. Fluids, 25, (2013), 23 p. 9

10 [19] Rionero, S. Soret effects on the onset of convection in rotating porous layers via the auxiliary system method. Ricerche di Matematica, Vol. 62, Issue 2, , (2013). [20] Merkin, D.R. Introduction to the theory of stability. Springer texts in Applied Mathematics, 24, (1997). [21] Gantmacher, F.R. The theory of matrices. Vol. 2, AMS (Chelsea Plublishing) (2000) [22] Rionero, S. Longtime behaviour of multicomponent fluid mixture in porous media. J. Eng. Sc., 48, , (2010). [23] Rionero, S. Symmetries and skew-symmetries against onset of convection in porous layers salted from above and below. Int. J. Nonlinear Mech., 47, 61-67, (2012). [24] Capone, F., Rionero, S. Inertia effect on the onset of convection in rotating porous layers via the auxiliary system method. Int. J. Non-linear Mech., 57, , (2013). 10

11 [25] Capone, F., De Luca, R. Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law. Int. J. of Non-Linear Mech. 47, issue 7, , (2012). [26] Capone, F., De Luca, R. Onset of convection for ternary fluid mixtures saturating horizontal porous layers with large pores. Rendiconti Lincei Matematica e Applicazioni, 23 (4), , (2012). [27] Rionero, S. Heat and Mass transfer by convection in multicomponent Navier-Stokes mixtures: absence of subcritical instabilities and global nonlinear stability via the Auxiliary System Method. Rend. Lincei Mat. Appl., 25, 1, 1-44, (2014). [28] Rionero, S. Convection in ternary porous layers with depthdependent permeability and viscosity. (submitted). 11

12 2 Discussion i) The heat and mass transfer by convection in ternary Brinkman porous layers with depth-dependent permeability and viscosities, via the Auxiliary System Method, is studied 2 ; ii) an own non linear system is obtained for the evolution of each Fourier component of the perturbations; iii) the linear instability captures completely the physics of the onset of convection since the absence of subcritical instabilities is shown together with the properties of linear stability to guarantee also the global non linear stability; iv) the thermal critical Rayleigh number is obtained in closed form; v) effects on the onset of convection in the earth s mantle, by virtue of depth-dependent permeability and viscosities, are 2 Further applications of the auxiliary system method can be found in [16], [21]-[28] 12

13 evaluated; vi) how large pores should be stratified in artificial porous materials in order to inhibit or promote the onset of convection, is analyzed. 13

14 3 Preliminaries Let Ox 1 x 2 x 3 be an orthogonal frame of reference with fundamental unit vectors e 1, e 2, e 3 (e 3 pointing vertically upwards). We assume that two different chemical components ( salts ) S α (α = 1, 2), have dissolved in the fluid and have concentrations C α (α = 1, 2), respectively, and that the equation of state is ] ρ = ρ 0 [1 α (T T 0 ) + A 1 (C 1 Ĉ1) + A 2 (C 2 Ĉ2), where ρ 0, T 0, Ĉα (α = 1, 2), are reference values of the density, temperature and salt concentrations, while the constants α, A α denote the thermal and solute S α expansion coefficients respectively (α = 1, 2). Combining Darcy s Law with (thermal) energy and mass balance together with the Boussinesq approximation, we obtain the fundamental equations governing the isochoric motions (when the pores x 3 = z [0, 1] are in the layer L] sufficiently large to generate 14

15 the Brinkman viscosity) [14]-[16] p= µ 1 K v gρ 0[1 α (T T 0 )+A 1 (C 1 Ĉ1)+A 2 (C 2 Ĉ2)] (µ 2 D ij )e i, x j i,j v = 0, T t + v T = k T, C 1t + v C 1 = k 1 C 1, C 2t + v C 2 = k 2 C 2, where (i = 1, 2) p= pressure field, µ 1 = f 1 (x 3 ) µ 1 = viscosity of the fluid, µ 2 = f 2 (x 3 ) µ 2 = viscosity of the fluid in the porous layer, K = Kf 3 (x 3 )= permeability, v= velocity, g= gravity, k= thermal diffusivity, K α = diffusivity of the solute S α, D ij = 1 2 ( vi x j + v j x i ), 15 (3.1)

16 K = constant reference value of permeability; µ i = constant reference value of viscosity µ i, (i = 1, 2). To (3.1) we append the boundary conditions T (0) = T 1, T (d) = T 2, C α (0) = C αl, C α (d) = C αu α = 1, 2, v e 3 = 0, at z = 0, d, (3.2) with T 1, T 2, C αl, C αu (α = 1, 2), positive constants. The boundary value problem (3.1)-(3.2) admits the conduction solution (ṽ, p, T, C α ) given by ṽ = 0, T = T1 βz, β = T 1 T 2, d C α = C αl z(δc α), C αl C αu = δc α, d[ P = p 0 + ρ 0 gz 2 α ] β 2 + A (δc 1 ) 1 2d + A (δc 2 ) 2 + [ 2d ] ρ 0 gz 1 α (T 1 T 0 ) + A 1 (C 1l Ĉ1) + A 2 (C 2l Ĉ2), 16 (3.3)

17 where p 0 is a constant. Setting v = ṽ + u, p = P + Π, T = T + θ, C α = C α + Φ α, (3.4) and introducing the scalings t = t d2 k, K u = u d, Π = Π µ 1k K, x = x d, θ = θ T, ( )1 Φ α = (Φ α ) Φ α, T µ1 k δt 2 ( = α ρ 0 g Kd, Φ µ1 kp α δc α α = A α ρ 0 g ( Kd α ρ 0 g R = Kd δt )1 2 ( Aα ρ 0 g, Rα = KdP )1 α δc α 2, µ 1 k ( ) µ 1 k D a = µ 2k µ 1 d 2, D ij = 1 u i 2 x + u j j x, i δt = T 1 T 2, H = sgn(δt ), H α = sgn(δc α ), P α = k k α, )1 2, (3.5) 17

18 since in the case at stake the layer is heated from below and salted from below by S 1 and from above by S 2, it follows that H = H 1 = 1, H 2 = 1 and the equations governing the dimensionless perturbations {u, Π, θ, (Φ α ) }, omitting the stars, are 1 3 Π= f 3 (x 3 )u+(rθ R 1 Φ 1 R 2 Φ 2 )e 3 +2D a u = 0, θ t + ( u θ = Ru e ) 3 + θ, Φ1 P 1 t + u Φ 1 = R 1 u e 3 + Φ 1, ( ) Φ2 P 2 t + u Φ 2 = R 2 u e 3 + Φ 2, under the boundary conditions i,j x j (gd ij )e i, (3.6) u e 3 = θ = Φ 1 = Φ 2 = 0 on z = 0, 1. (3.7) In (3.5)-(3.6) R and R α are the thermal and salt Rayleigh numbers 18

19 respectively while P α are the salt Prandtl numbers and D a is the Darcy number. We assume, as usually done in stability problems in layers, that i) the perturbations (u, v, w, θ, Φ 1, Φ 2 ) are periodic in the x and y directions, respectively of periods 2π/a x, 2π/a y ; ii) Ω = [0, 2π/a x ] [0, 2π/a y ] [0, 1] is the periodicity cell; iii) u, Φ 1, Φ 2, θ belong to W 2,2 (Ω) and are such that all their first derivatives and second spatial derivatives can be expanded in a Fourier series uniformly convergent in Ω and denote by by L 2(Ω) the set of functions Φ such that 1) Φ : (x, t) Ω R + Φ(x, t) R, Φ W 2,2 (Ω), t R +, Φ is periodic in the x and y directions of period 2π, 2π respectively a x a y and [Φ] z=0 = [Φ] z=1 = 0; 2) Φ, together with all the first derivatives and second spatial derivatives, can be expanded in a Fourier series absolutely uniformly 19

20 convergent in Ω, t R +. 4 The effective unknown fields This Section is devoted to show that (θ, Φ 1, Φ 2 ) are the effective unknown fields. Let us consider the b.v.p. 1 3 Π = f(z)u + (Rθ R 1 Φ 1 R 2 Φ 2 )e 3 + 2D a (gd ij )e i, x j u = 0, u e 3 = θ = Φ 1 = Φ 2 = 0, z = 0, 1. i,j (4.1) 20

21 Since the set {sin nπz} n N is a complete orthogonal system for L 2 (0, 1), then Φ {w, θ, Φ 1, Φ 2 } Φ = Φ n = Φ n (x, y, t) sin nπz. n=1 (4.2) By virtue of the periodicity in the x and y directions, one easily obtains 1 Φ n = a 2 Φ n, Φ n = ξ n Φ n, (4.3) with a 2 = a 2 x + a 2 y, Further, setting n=1 1 = 2 x y 2, ξ n = a 2 + n 2 π 2. (4.4) ζ = (rotu) e 3 = v x u y, (4.5) 21

22 from (4.1) 2 one obtains Since [ g(x3 ) 2 1 u = 2 w x z ζ y, ( ui + u )] j = 1 x j x i j 2 u j = ( u) = 0, x i x j x i it follows that 2(g(z)D ij ) j e i = g (z) g x j 1v = 2 w y z + ζ x. (4.6) ( ui + u ) ( j + 12 x j x g 2 u i i x 2 j g x j = 0, j = 1, 2, ( ) u z + w ) + u i, x i x j (4.7) + g(z) u (4.8) 22

23 and (4.1) becomes ( Π = f(z)u+ Setting Rθ ) 2 R α Φ α e 3 +D a [g (z)(u z + w) + g u]. α=1 (4.9) F = q(z)u, with q C 2 (0, 1), U = 0, (4.10) one easily obtains e 3 F = q U 3 z q U 3. (4.11) Applying (4.11) with F given respectively by f(z)u, g u z, g u, one obtains e 3 (fu) = f w z f w, e 3 (g u z ) = g w zz g w z, (4.12) e 3 (g u) = g w z g w. 23

24 Since e 3 (g w) = g 1 w, ( ) 2 e 3 Rθ R α Φ α = 1 (Rθ α=1 ) 2 R α Φ α, (4.13) the third component of the double curl of (4.1), is given by ) 2 A + B = 1 (Rθ R α Φ α, (4.14) with A = (f w z + f w), B = g w+2g w z +g (w zz 1 w). (4.15) For (w = w n, θ = θ n, Φ α = Φ αn ) one obtains ) 2 (A 1n + B 1n ) w n = a (Rθ 2 n R α Φ αn, (4.16) 24 α=1 α=1 α=1

25 with A 1n = (f z ) + f sin nπz, B 1n = g + 2g ( ) 2 z + g z 2 1 sin nπz, (4.17) (4.16), multiplied by sin nπz and integrated on (0, 1) gives ( ) 2 (A n + Bn) w n = a2 R θ n R α Φαn, (4.18) 2 with A n = Bn = α=1 ( nπ 2 f sin 2nπz + ξ n f sin 2 nπz) dz, {[ gξ 2 n + g ( ξ n + 2a 2)] sin 2 nπz g ξ n sin 2nπz } dz, (4.19) 25

26 i.e. with w n = Ãn ( Ã n = R θ n ) 2 R α Φαn, (4.20) α=1 a 2 2(A n + B n). (4.21) 5 Linear instability Setting L n = a 1n a 2n a 3n b 1n b 2n b 3n c 1n c 2n c 3n, (5.1) 26

27 a 1n = R 2 Ã n ξ n, a 2n = RR 1 Ã n, a 3n = RR 2 Ã n, b 1n = RR 1 Ã n, b 2n = R2 1Ãn + ξ n, b 3n = RR 1 Ã n, P 1 P 1 P 1 c 1n = RR 2 Ã n, c 2n = R 1R 2 Ã n, c 3n = R2 2Ãn ξ n, P 2 P 2 P 2 one easily obtains that θ n Φ 1n = t n=1 Φ 2n n=1 L n Linearizing it follows that θ n Φ 1n = L n t Φ 2n θ n Φ 1n Φ 2n θ n Φ 1n Φ 2n u n=1 θ n Φ 1n Φ 2n (5.2). (5.3), n {1, 2,...}. (5.4) 27

28 Denoting by (λ 1n, λ 2n, λ 3n ) the eigenvalues of L n, the spectral equation of L n is given by λ 3 in I 1nλ 2 in + I 2nλ in I 3n = 0, n {1, 2,...}, (5.5) with I 1n, I 2n, I 3n characteristic values (invariants) of L n [20]-[21], given by 3 I 1n = a 1n + b 2n + c 3n = λ αn, I 3n = detl n = λ 1n λ 2n λ 3n, α=1 a 1n a 2n I 2n = b 1n b 2n + a 1n a 3n c 1n c 3n + b 2n b 3n c 2n c 3n = λ 1n(λ 2n + λ 3n ) + λ 2n λ 3n (5.6) 28

29 and one easily obtains { [ R I 1n = R R2 2 + ξ )]} n (1 + 1P1 + 1P2 à n, P 1 P 2 à n I 2n = P [ 1 + P P1 + P 2 ξ n P 2 R P ] 1 R2 2 R 2 ξ n à n, P 1 P 2 P 1 + P 2 à n P 1 + P 2 P 1 + P 2 I 3n = 1 [ ( R 2 R1 2 R2 2 + ξ )] n à n ξ P 1 P n. 2 2 à n (5.7) By virtue of the Routh-Hurwitz conditions on the sign of the real parts of the eigenvalues of L n ([20], pp ), the following results hold: i) the conditions, (a 2, n) R + N, I 1n < 0, I 2n > 0, I 3n < 0, (5.8) are necessary for guaranteeing the linear asymptotic stability; 29

30 ii) if and only if, (a 2, n) R + N, I 1n < 0, I 3n < 0, I 1n I 2n I 3n < 0, (5.9) the thermal conduction solution is asymptotically linearly stable. Setting H = min H n, H n = ξ n, (5.10) (a 2,n) R + N Ã n ) R C1 = R2 1 R2 2 + (1 + 1P1 + 1P2 H, R C3 = R1 2 R2 2 + H, P 1 P 2 R C2 = 1 + P 2 R P ( ) 1 R H, P 1 + P 2 P 1 + P 2 P 1 + P 2 (5.11) in view of i) and ii) one obtains that iii) the conditions R 2 < R Cα, α = 1, 2, 3, (5.12) 30

31 are necessary for inhibiting the onset of convection; iv) if and only if R 2 < min(r C1, R C2 ), (R C1 R 2 )(R C2 R 2 ) > H (R C3 R 2 (5.13) ), P 1 + P 2 convection cannot occur and the thermal conduction solution is asymptotically linearly stable. Since (5.12) and (5.13) 1 are easily obtained, we confine ourselves to (5.13) 2. But it is easily verified that I 2n > I 3n is equivalent to I ( ) 1n 1 ξn P 2 R P 1 R2 2 > P 1 + P 2 Ã n P 1 + P 2 P 1 + P 2 ξ n /Ãn (R1 2 R2 2 + ξ n /Ãn) R 2 (5.14) P 1 + P 2 R1 2 ( R ), ξn R 2 P 1 P 2 P 1 P 2 Ã n which implies (5.13) 2. 31

32 6 Nonlinear stability via the Auxiliary System Method Setting X = X = θ n=1 Φ 1 Φ 2, F = u X n, X n = θ n Φ 1n Φ 2n θ Φ 1 Φ 2 = u X,, F n = u X n, F = (5.4) become X t = (L n X n + F n ), n=1 (X) t=0 = X (0) = X (0) n, X n = 0, on z = 0, 1, n=1 F n, n=1 (6.1) (6.2) 32

33 with X (0) n assigned. Following [17]-[19], we call auxiliary evolution system of the nth-fourier component of the perturbation X, associated to the velocity u given by (6.2), the system t X n = L n Xn + F n, X n = 0, on z = 0, 1, where F n and X n are given by F n = u X n, Xn = [ Xn ]t=0 = X(0) n, φ n φ 1n φ 2n (6.3). (6.4) Theorem 6.1 Let θ, Φ 1, Φ 2, φ n, φ 1n, φ 2n L 2(Ω) and let (u, θ, Φ 1, Φ 2 ) be solution of (6.2) and (φ n, φ 1n, φ 2n ) be solution, n N, of (6.3). Then the series φ n, φ in, (i = 1, 2) are a.e. con- n=1 n=1 33

34 vergent in Ω and it follows that φ n = θ, φ in = Φ i, (i = 1, 2). (6.5) n=1 Proof. Setting: S m = following i.b.v.p. holds S m m S 1m = t S 2m (S m ) t=0 = m n=1 n=1 n=1 m φ n, S im = n=1 L n φ n φ 1n φ 2n θ (0) n, (S im ) t=0 = m n=1 m φ in, (i = 1, 2), the n=1 u S m u S 1m u S 2m Φ (0) in, (i = 1, 2),, (6.6) S m = S im = φ n = φ in = 0, z = 0, 1, i = 1, 2 n = 1,...m. (6.7) 34

35 Setting Ψ n = Ψ in = θ n φ n, for n = 1, 2,...m, θ n, for n > m, Φ in φ in, for n = 1, 2,...m, Φ in, for n > m,, Ψ =, Ψ i = by virtue of (6.3) and (6.6)-(6.7), one obtains, Ψ Ψ n u Ψ Ψ 1 = L n Ψ 1n u Ψ 1 t Ψ 2 n=1 Ψ 2n u Ψ 2 under the i.b.c. (α = 1, 2) (Ψ) t=0 = θ n (0), (Ψ i ) t=0 = n=m+1 n=m+1 35 Ψ n (6.8) n=1 Ψ in (6.9) n=1, (6.10) Ψ (0) in ; Ψ = Ψ α = 0, z = 0, 1. (6.11)

36 Since lim m n=m+1 θ (0) n = lim m n=m+1 Φ (0) in i.b.v. admits only the null solution, it follows that = 0 and (6.10) under zero lim (θ S m) = lim (Φ i S im ) = 0, (i = 1, 2). (6.12) m m Theorem 6.2 Let (5.13) hold. Then the zero solution of (6.2) is globally asymptotically stable, i.e. the thermal conduction solution is linearly stable and non linearly globally asymptotically stable with respect to the L 2 (Ω)-norm. Proof. The proof can be obtained following step by step the proof of theorem 7.1 given in [17] in the absence of depth-dependence permeability and viscosities. 36

37 7 Global non linear stability via symmetries and skewsymmetries hidden in (3.6) Setting Ψ 1 = R 1 θ P 1 RΦ 1, Ψ 2 = R 2 θ + P 2 RΦ 2 ; it follows that (3.6) is equivalent to (α = 1, 2) Π = f(z)u + 1 ( R θ + R 1 Ψ 1 R ) Ψ 2 e 3 + 2D a (gd ij )e i, R P 1 P 2 x j u = 0, dθ dt = Rw + θ, P under the boundary conditions with dψ α α dt i,j = Ψ α + R α (P α 1) θ, (7.1) w = θ = Ψ α = 0, on z = 0, 1, (7.2) R = R 2 R2 1 P 1 + R2 2 P 2. (7.3) 37

38 System (7.1) 1 -(7.2) 2 together with (7.3) is reducible to (3.6) via the substitution R R 1 R P 1 R R 2 P 2 R Ψ 1 Ψ 2 (7.4) R R 1 R 2 Φ 1 Φ 2 and therefore it follows that θ θ n Ψ 1 = L n Ψ 1n t Ψ 2 n=1 Ψ 2n u θ Ψ 1 Ψ 2, (7.5) with R R 1 à n ξ n à n R 2 à n P 1 P 2 L n = R 1 (P 1 1)ξ n ξ n 0. (7.6) P 1 P 1 R 2 (P 2 1)ξ n 0 ξ n P 2 P 2 Theorem 7.1 The global non linear stability of the conduction 38

39 solution is guaranteed by R 2 < R 2 1 R H = R C3, for P 1 1, P 2 1, (7.7) R 2 < R2 1 P 1 R2 2 P 2 + H, for P 1 1, P 2 1, (7.8) R 2 < R 2 1 R2 2 P 2 + H, for P 1 1, P 2 1, (7.9) R 2 < R2 1 P 1 R H, for P 1 1, P 2 1, (7.10) where H is given by (5.10). Proof. Since the operator L n can be obtained by the operator L n appearing in (6.6) of [23] by putting Ãn at the place of η n, following step by step the proof given in [23], the theorem immediately follows. For the sake of completeness, we give here a sketch of the proof of (7.7). By virtue of the absence of subcritical instabilities, (7.5) 39

40 reduces to Setting t ψ αn = θ n ψ 1n ψ 2n = L n (7.11) for (P 1 1, P 2 1) becomes θ n ψ 1n t ψ 2n θ n ψ 1n ψ 2n. (7.11) P α 1 ξ n à n φ αn, (α = 1, 2) (7.12) = L n θ n ψ 1n ψ 2n (7.13) 40

41 L n = with L n symmetric operator given by R à n ξ n R 1 P 1 (1 P 1 )ξ n à n R 2 P 2 (P 2 1)ξ n à n R 1 (1 P 1 )ξ n à n ξ n 0 P 1 P 1 R 2 P 2 (P 2 1)ξ n à n 0 ξ n P 2 (7.14) Since, n N, the eigenvalues of L n are real numbers, the marginal state is a stationary state and the critical Rayleigh number is given by the invariant R C3 and, in view of (5.12)for α = 3, (7.7) immediately follows. 8 Applications By virtue of (5.11)-(5.13) and (7.7)-(7.10), the influence of depthdependence on the onset of convection, is measured via H : the onset of convection is delayed by the increasing of H. We now, for 41.

42 the sake of simplicity, confine ourselves to the case g = 1 + bz, g 0 in (0, 1), (8.1) with b real constant. In view of (8.1) and 1 0 g sin 2nπz dz = b nπ 1 0 d dz ( sin 2 nπz ) dz = 0, (8.2) it follows that 1 ( A n = n 2 π 2 cos 2 nπz + a 2 sin 2 nπz ) f dz, 0 1 Bn = ξn 2 g sin 2 nπz dz, 0 H n = 2ξ n = 2 1 ( ) Ã n a 2ξ n (A n + Bn) > 2ξ n f + ξ2 n 0 a 2g sin 2 nπz dz. (8.3) Therefore H = min H n = min H 1. (8.4) (a 2,n) R + N a 2 R + 42

43 In view of lim H 1 = lim H 1 =, (8.5) a 2 0 a 2 it follows that there exists a positive bounded critical value a 2 c > 0 such that H = (H 1 ) a 2 =a 2. (8.6) c Setting h 1 = 1 0 it easily follows that f sin 2 πz dz, h 2 = 1 0 f cos 2 πzdz, h 3 = B 1 a 2 + π 2, (8.7) H 1 a = 0 2 π2 (a 2 h 1 + π 2 h 2 ) = a 2 (a 2 + π 2 )(h 1 + h 3 ) (8.8) and hence a 2 π 2 } c = {[h h 2 (h 1 + h 3 )] 1 2 h 3. (8.9) 2(h 1 + h 3 ) 43

44 Denoting by Hu and Hl an upper and a lower bound of H, in view of 1 1 sin 2 πz dz = cos 2 πz dz = , 1 g sin 2 πz dz = 1 1 (8.10) 2 + b z sin 2 πz dz, 0 one obtains Hl = 2 [ (π (a 2 a 2 c + π a 2 ) c) f + (a 2 c + π 2 ) 2ḡ ] > c 2 2 > (π 2 + a 2 c)[ f + (a 2 c + π 2 )ḡ] > π 2 ( f + π 2 ḡ), Hu = (a2 c + π 2 ) 2 [ f + (a 2 c + π 2 )ḡ], a 2 c 0 (8.11) with f = ess inf (0,1) f, f = ess sup(0,1) f, ḡ = ess infg > 0, ḡ = ess sup (0,1) g. 7.1 Applications to the earth s mantle The increase in viscosity µ 1 in earth s mantle proposed by Torrance 44

45 and Turgotte (see [1], p.118) is given by µ 1 = µ 1 f 1 (z), f 1 (z) = e c(1/2 z), c = const. > 0. (8.12) For a linear variation in the permeability given by K = Kf 3 (z), f 3 (z) = 1 + γz, γ = const., (8.13) one obtains f = ec(1/2 z) 1 + γz. (8.14) The corresponding values of H, for various values of c, γ, b are furnished by the following table. 7.2 Stratification of large pores in artificial porous materials The stratification of pores in artificial porous materials has a very relevant interest. In fact, for instance, in the construction of porous materials for insulating purposes, the aim is to delay or prohibit heat transfer and hence high thermal critical Rayleigh numbers are requested. Viceversa in cooling pipes used in computers or in other modern devices, the porosity has to be stratified in such a way to 45

46 Table 1: Values of H for the earth s mantle f g h 1 h 2 h 3 a c H e (0.5 z) z e (0.5 z) 1 0.5z e 0.5(0.5 z) 1 0.2z e 0.5(0.5 z) z e 0.5(0.5 z) 1 0.2z e 2.5(0.5 z) 1 0.8z e (0.5 z) z e 0.5(0.5 z) 1 + 2z 1+0.5z z z z z z z z produce low thermal critical Rayleigh numbers. We now confine ourselves to the stratification of large pores in porous artificial materials assuming f = 1 and ḡ given by (8.1). Values of g guaranteeing increasing values of H and hence high 46

47 thermal critical Rayleigh numbers are furnished in table 2. Table 2: Stratification of large pores for prohibiting heat transfer f g h 1 = h 2 h 3 a c H z 1/ z 1/ z 1/ z 1/2 3/ z 1/ z 1/ z 1/ z 1/ z 1/ z 1/ z 1/

48 Values of g guaranteeing decreasing values of H and hence low thermal critical Rayleigh numbers are furnished in table 3. Table 3: Stratification of large pores for rapid heat transfer f g h 1 = h 2 h 3 a c H z 1/ /5z 1/ /3z 1/ /2z 1/ /3z 1/ /4z 1/ /8z 1/ /8z 1/